Question

use the table from problem 1 to find the following probabilities. write answers as fractions. example: if the fraction is two - thirds, type in 2/3
a) the probability that a woman tests positive given that she has breast cancer 7/8
b) the probability that a woman tests positive given that she does not have breast cancer 70/992
determine where or not the events of having breast cancer and testing positive for breast cancer are independent. show all relevant calculations. write answers as fractions.
$p(a\text{ and }b)=p(a)\cdot p(b)$(used to determine independence of events)
c) what is the probability of having breast cancer? type your answer...
d) what is the probability of testing positive for breast cancer? type your answer...
e) what is the probability of having breast cancer and testing positive? type your answer...
f) are the events of having breast cancer and testing positive for breast cancer independent? use the work from problems c, d, and e to help you answer.
type in yes or no. type your answer...
g) find the probability that a woman has breast cancer given that her test result is positive. type your answer...
h) find the probability that a woman does not have breast cancer given that her test result is negative. type your answer...

Explanation:

Step1: Recall the table from Problem 1 (assuming standard breast cancer screening table, e.g., total women = 1000, breast cancer cases = 8, non - cancer = 992)

To find the probability of having breast cancer, we use the formula \(P(A)=\frac{\text{Number of women with breast cancer}}{\text{Total number of women}}\).
Assuming the total number of women is 1000 (a common total in such problems) and the number of women with breast cancer is 8 (from standard tables or previous problem context).

Step2: Calculate the probability

So \(P(\text{having breast cancer})=\frac{8}{1000}=\frac{1}{125}\) (simplifying the fraction by dividing numerator and denominator by 8: \(8\div8 = 1\), \(1000\div8=125\)).

Answer:

\(\frac{1}{125}\)